Econometrics Example Sheet

Problem 1:
From the data for 46 states in the US for 1992, Baltagi obtained the following regression results:
LogC(hat)= 4.3 1.34 log(P) + 0.17 log(Y)
se
(0.91) (0.32)
(0.2)
Adjusted R2=0.27
where C is Cigarette consumption (packs per year)
P is real price per pack
and Y is real disposable income per capital
1. Interprete the meaning of each estimated coefficients
2. Are the signs of coefficients of your expectation
3. How much P and Y explain for the variation in C
4. What is the elasticity of demand for cigarette w.r.t. price? Is it statistically significant? If so,
is it statistically different from 1?
5. What is the income elasticity of demand for cigarette? is it statistically significant? If not,
what might be the reasons for that?
6. Test for overall significance of the model
Problem 2:
You want to study the dependence of beer expenditures of employees in a company on their
incomes, ages and sexes. You have collected a random sample of observations on 40 office
employees, 20 of whom are females and 20 of whom are males. Here is the description of variables
in the data set:
BEi :
INCi :
AGEi :
SEXi :
male.

the annual beer expenditures of employee i, measured in dollars per year;

the annual income of employee i, in thousands of dollars per year;
the age of employee i, in years;
the dummy variable, SEXi = 1 if employee i is female and SEXi = 0 if employee i is

You propose the following model (model (1)):

BEi 1 2 INC i 3 AGE i 4 SEX i 5 SEX i * INC i 6 SEX i * AGE i u i

Using OLS method in EVIEWS, you obtain the following results:

Result (1)
Dependent variable: BE
Included observations: 40
Variable
Coefficient
C
489.8631
INC
0.002893
AGE
-10.07924
SEX
-265.8574
SEX*INC
-0.001029
SEX*AGE
4.231494

Std. Error
73.85524
0.000775
2.229676
113.3658
0.000971
3.648383

t-Statistic
6.632747
3.734180
-4.520493
-2.345129
-1.059491
1.159827

Prob.
0.0000
0.0007
0.0001
0.0250
0.2968
0.2542

R-squared

0.6470

Result (2)
Ramsey RESET Test:
F-statistic
Log likelihood ratio

2.110154
7.432899

Probability
Probability

0.119102
0.059308

Breusch-Godfrey Serial Correlation LM Test:

F-statistic
0.545784 Probability
Obs*R-squared
1.319452 Probability

0.584685
0.516993

Result (3)

Result (4)
White Heteroskedasticity Test:
F-statistic
0.768684
Obs*R-squared
7.495667

Probability
Probability

0.645556
0.585656

Result (5)
BEi = 459.21+ 0.0023 INCi - 8.42 AGEi -169.87 SEXi

R2=0.6294

Result (6)
BEi = 342.88+ 0.00238 INCi - 7.575 AGEi

R2= 0.3292

1. Write down the sample regression model of model (1) based on the result (1)? Write down the
population regression model and sample regression model for male and female employees and
explain the meaning of the estimated regression coefficients?
2. Use results (2), (3) and (4) to test for possible problems in the estimated model of model (1). In
each test, specify clearly type of test, type of problem, the statistic used, null and alternative
hypothesis and conclusion about the problem.
3. Using result (1), for male employees, how the expenditures for beer change if their income
increases 1000USD/year? Answer the same question for female employees given that:
cov( 2 , 5 ) 0

4. In the model (1), state the null and the alternative hypothesis if you want to test that the models
for the expenditures of beer for male and female are not different in slope coefficients of both INC
and
AGE. In other words, you want to conduct the joint test of hypothesis of equal slope
coefficients of male and female for INC and equal slope coefficients of male and female for AGE.
Perform this test using appropriate information given above.
5. Using the results above to test the hypothesis that the variable SEX does not affect the annual
expenditures for beer.
Problem 3:
A researcher is using data for a sample of 526 paid workers to investigate the relationship between
hourly wage rates Yi (measured in dollars per hour) and years of formal
education Xi(measured in years). Preliminary analysis of the sample data produces the following
sample information:
N = 526 ; Yi =3101.35; Xi =6608; Yi2= 25446.29 ; Xi2= 87040

Xi Yi= 41140.65; xi yi=2179.204; yi2=7160.414; xi2= 4025.43; u2=5980.682

1. Use the above information to compute OLS estimates of the intercept coefficient and slope
coefficient?
2. Interpret the slope coefficient estimate you calculated in part 1
3. Calculate an estimate of 2, the error variance
4. Compute the value of r2, the coefficient of determination for the estimated OLS sample
regression equation. Briefly explain what the calculated value of r2 means?
5. Test the opinion that years of formal education does not affect hourly wage rates
6. Compute two-side 95% CI for slope coefficient. Would two-side 99% CI be wider or
narrower than two-side 95% CI and why?
7. There is opinion that when formal education increase 1 year, the average wage rates
increase at lease 0.5 USD/h. Test this opinion.
8. Predict the average Y when X is 10 years.
Problem 4:

Dependent Variable: GNP

Included observations: 15
Variable
Coefficient
C
-529.6074
CPI
14.18311
I
1.325414
R
6.541530
R-squared
0.999646
Adjusted R-squared 0.999550
S.E. of regression
15.65737
Sum squared resid
2696.686
Log likelihood
-60.22204
Durbin-Watson stat
1.764303
1.
2.
3.
4.
5.
6.
7.
8.

Std. Error
t-Statistic
18.04051 -29.35656
0.382173
37.11174
0.123177
10.76025
3.058294
2.138947
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)

Prob.
0.0000
0.0000
0.0000
0.0557
1748.647
738.1458
8.562939
8.751752
10368.12
0.000000

Read the information from the above report.

Write down SRF and explain the meaning of each estimated coefficients.
Test for significance of each variable
Test for the significance of all independent variables simultaneously.
How much the independent variables can explain for the variation of GNP.
When investment increases 1bil, in which range can GNP increase?
Test for hypothesis that GNP will increase at least 15 bils if CPI increase 1%.
Test for the hypothesis that CPI and I have the same effects on GNP given that the
covariance between two corresponding estimated coefficients is .04228

Problem 5
The demand for roses was estimated using quarterly figures for the period 1971 (3rd quarter) to 1975
(2nd quarter). Two models were estimated and the following results were obtained:
Y = Quantity of roses sold (dozens)
X2 = Average wholesale price of roses ($ per dozen)
X3 = Average wholesale price of carnations ($ per dozen)

X4 = Average weekly family disposable income ($ per week)

X5 = Time (1971.3 = 1 and 1975.2 = 16)
ln = natural logarithm
The standard errors are given in parentheses.
A.
ln Yt = 0.627 - 1.273 ln X2t + 0.937 ln X3t + 1.713 ln X4t - 0.182 ln X5t
(0.327)
(0.659)
(1.201)
(0.128)
R2 = 77.8%
D.W. = 1.78
N = 16
B.

ln Yt = 10.462 - 1.39 ln X2t

(0.307)
R2 = 59.5%
D.W. = 1.495

N = 16

Correlation matrix:

ln X2

ln X2

ln X3

1.0000

-.7219

ln X4

ln X5

.316

-.7792

0
ln X3

-.7219

1.0000

-.1716

.5521

ln X4

.3160

-.1716

1.0000

-.6765

ln X5

-.7792

.5521

-.6765

1.0000

a) How would you interpret the coefficients of ln X2, ln X3 and ln X4 in model A?

What sign would you expect these coefficients to have? Do the results concur with your expectation?
b) Are these coefficients statistically significant?
c) Use the results of Model A to test the following hypotheses:
i) The demand for roses is price elastic
ii) Carnations are substitute goods for roses
iii) Roses are a luxury good (demand increases more than proportionally as income rises)
d) Are the results of (b) and (c) in accordance with your expectations? If any of the tests are
statistically insignificant, give a suggestion as to what may be the reason.
e) Do you detect the presence of multicollinearity in the data? Explain.
f) Do you detect the presence of serial correlation? Explain
g) Do the variables X3, X4 and X5 contribute significantly to the analysis? Test the joint significance of
these variables.
h) Starting from model B, assuming that at the time point of January, 1973, there was a disaster that
heavily affected the quantity of roses produced. Suggest a model to check if we have to use two
different models for the data before and after the disaster. (Using dummy variable).
Problem 6:
Two large US corporations, General Electric and Westinghouse, compete with each other and
produce many similar products. In order to investigate whether they have similar investment
strategies, we estimate the following model using pooled time series data for the period 1935 to
1954 for the two firms:

INVt = 1 + 2DVt + 3Vt + 4DV*Vt + 5Kt + 6DV*Kt + ut

where

(1)

INV = gross investment in plant and equipment

V = value of the firm = value of common and preferred stock
K = stock of capital
DV = 0 if General Electric (observations 1 to 20)
= 1 if Westinghouse (observations 21 to 40)

All three continuous variables are measured in millions of 1947 dollars. Pooling the data yields 40
observations with which to estimate the parameters of the investment function. However, pooling
is valid only if the regression parameters are the same for both firms. In order to test this
hypothesis, intercept and slope dummy variables are included in the model.
Dependent Variable: INV
Method: Least Squares
Sample: 1 40
Included observations: 40
Variable
Coefficien
t
C
-9.956306
DV
9.446916
V
0.026551
DV*V
0.026343
K
0.151694
DV*K
-0.059287
R-squared
0.827840
Adjusted R-squared 0.802523
S.E. of regression
20.99707
Sum squared resid
14989.82
Log likelihood
-175.2825
Durbin-Watson stat 1.121571

Std. Error

t-Statistic

23.62636 -0.421407
28.80535 0.327957
0.011722 2.265064
0.034353 0.766838
0.019356 7.836865
0.116946 -0.506962
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)

Prob.
0.6761
0.7450
0.0300
0.4485
0.0000
0.6155
72.59075
47.24981
9.064124
9.317456
32.69818
0.000000

(a) Interpret all the coefficient estimates, stating whether the signs are as you would expect, and
comment on the statistical significance of the individual coefficients.
(b) Comment on the overall fit and statistical significance of the model.
(c) The Jarque-Bera statistic is 7.77 and its p-value is 0.02. What can you conclude about the
distribution of the disturbance term? Why is this test important?
(d) On the basis of the above results, is pooling the data from the two firms appropriate? Explain.
(e) An alternative way of testing whether pooling the data is appropriate, without using dummy
variables, is to use the Chow breakpoint test. Referring to table below, briefly discuss how the
test works and whether the results are consistent with the earlier model (which includes
dummy variables).

Chow Breakpoint Test: 21

F-statistic
1.189433
Log likelihood ratio 3.992003

Probability
Probability

0.328351
0.262329

(f) Explain the results and implications of the following Ramsey RESET test. (Note that the
dummy variables have been omitted from the original model).
Ramsey RESET Test:
F-statistic
0.000200
Log likelihood ratio 0.000219
Test Equation:
Dependent Variable: INV
Method: Least Squares
Date: 05/15/02 Time: 13:07
Sample: 1 40
Included observations: 40
Variable
Coefficien
t
C
17.81458
V
0.015226
K
0.144467
FITTED^2
-2.87E-05
R-squared
0.809773
Adjusted R-squared 0.793921
S.E. of regression
21.44950
Sum squared resid
16562.91
Log likelihood
-177.2784
Durbin-Watson stat 1.106556

Probability
Probability

Std. Error

0.988806
0.988189

t-Statistic

8.199161 2.172732
0.006706 2.270632
0.065596 2.202383
0.002028 -0.014128
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)

Prob.
0.0365
0.0293
0.0341
0.9888
72.59075
47.24981
9.063919
9.232807
51.08255
0.000000

Note: We can have similar questions using results from eviews to check for autocorrelation and
heteroscedasticity (Breusch Godfrey test and White test).